Since you are in the Los Angeles area, the logical place to take the gun for a quick once-over as to condition, value, and whether you should do anything other than clean it would be Jim Thynne at Ivory Beads in West Covina. Jim is probably the most knowledgeable Parker guy in the area, and you can trust his advice. The shop will also have at least 20 other Parkers at any given time that you can compare yours to, just for reference. Ivory Beads is a retail operation and does not do any gunsmithing. Jim can recommend smiths for any inspection, cleaning, or repairs, or can arrange to have those services performed for you.
Freddy Brunner in Escondido is a solid choice for work on this gun, and is less tha two hours away from most of LA. Steve Matthews is the resident gunsmith at Bain & Davis Sporting Goods in San Gabriel, and knows his way around old doubles. Freddy and Steve have both been entrusted with some of my favorite guns.
Regarding the immediate cleaning and oiling, WD-40 is OK for a field wipe down on a rainy day, but I'd rather see you use Rem Oil or CLP, or even 3-in-1 Oil, available at any hardware store. Parker originally recommended 3-in-1 for guns of your vintage, and I still use it. Walt Schiessl (The L.C. Smith Man) recommends Singer Sewing Machine Oil for Elsies because it's easy on the wood while protecting the metal. Put a dab of good grease on the hinge pin or barrel hook. Once you have had the gun inspected and the internals cleaned and lubed (very important on a gun of this age), you should be able to shoot it with some degree of impunity. Listen to your gunsmith.